Final answer:
This answer provides a thorough explanation of various aspects of the given elliptic curve, including its properties, points over real and integer numbers, point doubling and addition, and the bound for the number of points using Hesse's theorem.
Step-by-step explanation:
(i) Confirm that it is an elliptic curve: To confirm that the curve y^2 ≡ x^3 + 3x + 5 mod 17 is an elliptic curve, we need to show that it satisfies the required properties. These properties include the non-singularity of the equation, the smoothness of the curve, and the existence of a rational point.
(ii) Determine three points on the curve over real numbers: To determine three points on the curve over real numbers, we can substitute different values of x into the equation and solve for y. For example, when x = 1, we get y = ±4 mod 17, giving us two points (1, 4) and (1, -4).
(iii) Determine three points on the curve over integer numbers: To determine three points on the curve over integer numbers, we can substitute different values of x into the equation and solve for y. For example, when x = 0, we get y = ±2 mod 17, giving us two points (0, 2) and (0, -2).
(iv) For one of the points P in (iii), find 2P (or double): To find 2P, we can use the point doubling formula. For example, if we choose the point (0, 2), we can substitute its coordinates into the formula and solve for the coordinates of 2P.
(v) For two of the points P and Q in (iii), find P+Q: To find P+Q, we can use the point addition formula. For example, if we choose the points (0, 2) and (1, 4), we can substitute their coordinates into the formula and solve for the coordinates of P+Q.
(vi) Find the bound for the number of points on this curve using Hesse’s theorem: To find the bound for the number of points on this curve, we can use Hesse's theorem. This theorem provides an upper bound for the number of rational points on an elliptic curve based on the coefficients of its defining equation.